Standard bases in mixed power series and polynomial rings over rings. (March 2017) - Markwig, Thomas; Ren, Yue; Wienand, Oliver
Abstract: In this paper we study standard bases for submodules of a mixed power series and polynomial ring R 〚 t 1, …, t m 〛 [ x 1, …, x n ] s respectively of their localisation with respect to a t _ -local monomial ordering for a certain class of noetherian rings R, also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora and to generalise the Buchberger criterion. Everything else then translates naturally. Setting either m = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings over R as a special case.